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Crystal Structure and Dynamics Paolo G Crystal Structure and Dynamics Paolo G. Radaelli, Michaelmas Term 2012 Part 3: Dynamics and phase transitions Lectures 8-10 Web Site: http://www2.physics.ox.ac.uk/students/course-materials/c3-condensed-matter-major-option Bibliography ◦ M.S. Dresselhaus, G. Dresselhaus and A. Jorio, Group Theory - Application to the Physics of Condensed Matter, Springer-Verlag Berlin Heidelberg (2008). A recent book on irre- ducible representation theory and its application to a variety of physical problems. It should also be available on line free of charge from Oxford accounts. ◦ Volker Heine Group Theory in Quantum Mechanics, Dover Publication Press, 1993. A very popular book on the applications of group theory to quantum mechanics. ◦ Neil W. Ashcroft and N. David Mermin, Solid State Physics, HRW International Editions, CBS Publishing Asia Ltd (1976). Among many other things, it contains an excellent de- scription of thermal expansion in insulators. ◦ L.D. Landau, E.M. Lifshitz, et al. Statistical Physics: Volume 5 (Course of Theoretical Physics), 3rd edition, Butterworth-Heinemann, Oxford, Boston, Johannesburg, Melbourn, New Delhi, Singapore, 1980. Part of the classic series on theoretical physics. Definitely worth learning from the old masters. 1 Contents 1 Lecture 8 — Lattice modes and their symmetry 3 1.1 Molecular vibrations — mode decomposition . .3 1.2 Molecular vibrations — symmetry of the normal modes . .8 1.3 Extended lattices: phonons and the Bloch theorem . .9 1.3.1 Classical/Quantum analogy . 11 1.3.2 Inversion and parity . 12 1.4 Experimental techniques using light as a probe: “Infra-Red” and “Raman” . 12 1.4.1 IR absorption and reflection . 13 1.5 Raman scattering . 15 1.6 Inelastic neutron scattering . 16 2 Lecture 9 — Anharmonic effects in crystals 18 2.1 Thermal expansion . 18 2.1.1 Thermal expansion in metals . 21 2.2 Conservation of crystal momentum . 21 2.3 Heat transport theory . 23 2.4 Thermal conductivity due to phonons . 25 3 Lecture 10 — Phase Transitions 29 3.1 Continuous and discontinuous phase transitions . 29 3.2 Phase transitions as a result of symmetry breaking . 29 3.3 Macroscopic quantities: the Neumann principle . 30 3.3.1 Polarisation and ferroelectricity . 31 3.3.2 Magnetisation and ferromagnetism . 31 3.4 The Landau theory of phase transitions . 31 3.5 Analysis of a simple Landau free energy . 34 3.5.1 The order parameter (generalised polarisation) . 34 3.5.2 The generalised susceptibility . 36 3.5.3 The specific heat . 37 3.6 Displacive transitions and soft modes . 37 2 1 Lecture 8 — Lattice modes and their symmetry 1.1 Molecular vibrations — mode decomposition • We have seen in previous sections that dynamical effects can destroy the translational symmetry in a crystal, giving rise to scattering outside the RL nodes. In addition, other symmetries will be broken by individual excitations. Here, we want to illustrate how symmetry breaking can be classified with the help of symmetry. • Normal modes of vibration can be fully classified based on the symmetry of the potential. This is an entirely classical derivation. • In the quantum realm, the eigenstates of a Hamiltonian with a given sym- metry will not possess the full symmetry of the Hamiltonian, but can also be classified on the basis of symmetry. This is one of the most power- ful applications of symmetry to quantum mechanics: one can deduce the whole multiplet structure of a Hamiltonian from symmetry consider- ations alone. • Since the presence of a lattice gives rise to additional complications, we will first illustrate the principle using isolated molecules. We will show that molecular vibrations can break the symmetry of the molecule in a systematic way. • Isolated molecules possess a point-group symmetry, which is not restricted to be one of the 32 crystallographic point groups. • In the remainder, we will call modes the static patterns of distortion of a molecule, which can be thought of as snapshots of the molecule as it vibrates (strictly speaking, displacements in a mode can have complex coefficients — see below). Normal modes will have the usual meaning of special patterns of distortions associated with a single vibrational frequency !. An example of a mode is given in fig. 1 for a hypothetical square molecule. • The mode in fig. 1 has no symmetry whatsoever. However, one can con- struct modes that retain some of the original symmetry. Modes in fig. 2, for example, are not completely arbitrary: they transform in a well- defined way by application of the symmetry transformations of the origi- nal molecule (which has point-group symmetry 4mm in 2D). Specifically, they are either symmetric or antisymmetric upon application of any of the 8 symmetry operators of 4mm (see Lecture 1): 3 Figure 1: A snapshot of a vibrating square molecule. Mode Γ1 is symmetric under all the symmetry operators of the group — we say that it transforms under the totally symmetric mode. Mode Γ2 is symmetric under 1, 2, m10 and m01 and antisymmetric + − under 4 , 4 , m11 and m11¯. Mode Γ3 is symmetric under 1, 2, m11 and m11¯ and antisymmetric + − under 4 , 4 , m10 and m01. + − Mode Γ4 is symmetric under 1, 2, 4 and 4 and antisymmetric under m10, m01, m11 and m11¯. • Once can also say that each “symmetric” operator is equivalent to mul- tiplying all the displacements of a given mode by +1, whereas each “antisymmetric” operator by −1. This is the simplest example of a irre- ducible representation of a group — a central concept in group the- ory: the action of a symmetry operator on a mode has been “reduced” to multiplying that mode by a number. In a concise way, we could write, for example: 4+ [Γ2] = −1 [Γ2] 2 [Γ3] = +1 [Γ3] m11 [Γ4] = −1 [Γ4] (1) • Not all modes can be fully “reduced” in this way. An example is given in fig. 3 . We can see that: 4 Figure 2: The four “1 dimensional modes” of the square molecule. These modes transform into either themselves (symmetric) or minus themselves (antisymmetric) upon all symmetries of the molecule. Figure 3: The four “2 dimensional modes” of the square molecule. These modes transform into either ± themselves (symmetric/antisymmetric) or into each other in pairs upon all symmetries of the molecule. Note that all these modes are antisymmetric upon 2-fold rotation. 5 Certain symmetry operators interchange the modes. For example, the operator 4+ transforms mode [I] into mode [II] and [III] into mode IV ], etc. One can prove that there is no way of decomposing these modes as a linear combination of ”fully reduced” modes that transform as the previous group, i.e., as a multiplication by +1 or −1. [I] is never transformed into [III] (or vice versa) and [III] is never transformed into [IV ] (or vice versa). • Here, it is clearly impossible to transform these modes by multiplying each of them by a number. However, the symmetry operations on these modes can still be summarised in an extremely concise mathematical form. In order to achieve this, we can consider these modes as ba- sis vectors of an abstract mode space. Linear combinations of these modes simply mean vector addition of the displacements of each atom. • Crucially, the symmetry transformations preserve the linearity of mode space, so that, if g is an operator, m1 and m2 are modes and a and b are constants. This is a key requirement of a representation of the group. g[am1 + bm2] = agm1 + bgm2 (2) • Let us consider in particular the set of displacements that are linear com- binations of modes [I] and [II] — in other words, all the displacements of the type a a[I] + b[II] ! (3) b where the array notation in eq. 3 should be obvious. An alternative phrasing is that we are considering the subspace spanned by modes [I] and [II]. • The transformations can now be expressed in matrix form, as illustrated in tab. 1. Modes [III]-[IV ] transform in the same way. • As a second example, we analyse the displacements of the central atom of our hypothetical molecule, located on the fourfold axis. This atom has two degrees of freedom, as shown in fig 4. The two correspond- ing modes transform in the same way as modes [I] and [II] (or [III] and [IV ]). In the language of representation theory, we say that there transform according to irreducible representation Γ5. 6 Table 1: Matrix representation of the transformations of point group 4mm on the subspace spanned by modes [I] and [II] 1 2 4+ 4− 1 0 −1 0 0 −1 0 1 0 1 0 −1 1 0 −1 0 m10 m01 m11 m11¯ 1 0 −1 0 0 1 0 −1 0 −1 0 1 1 0 −1 0 Figure 4: The two central-atom modes of the square molecule. One can verify that they transform as the ”2-D” corner modes, i.e., with the representation Γ5 • The 10 modes described here above exhaust all the 8 degrees of freedom of the 4 atoms at the corner of the molecule plus the atom at the centre. Therefore, any arbitrary displacement pattern can be written as a linear combination of the 10 modes, which represent a complete basis for the space of all possible distortions of the molecule. The theory of irreducible representations teaches how to decompose arbitrary displacement patters as linear combinations of special basis modes, transforming according to rules similar to the ones illustrated here above. 7 1.2 Molecular vibrations — symmetry of the normal modes • We will now see how the analysis in the previous section helps us to iden- tify normal modes without knowing anything about the potential (other than its symmetry). Let us consider a normal mode Qi.
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